The Geometric Distribution: A \"discrete\" distribution In independent trials, only two outcomes are possible: \"success\" and \"failure.\" For every trial, the probability of \"success\" is p (0,1). The trial on which the first success occurs is a random variable with a \"geometric\" distribution; the probability that the first success occurs at trial x is: Pr(X = x) = f(x) = p (1 - p)x-1, x = 1, 2, 3, ... p (0,1) is the probability of \"success\"; 1 - p probability of \"failure.\" Since 0 < p < 1, 1 - p will also be between 0 and 1. p = 0.56 p (0,1) D p = 0.02 The geometric distribution Cumulative distribution Probability: Pr(X = x) 1.0 0.8 Pr(X = x) 0.6 Welcome to Professor Mitchell's interactive geometric distribution worksheet! 0.4 x1 = 2 x1 Z++ x2 = 3 x2 Z++ 0.2 0.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3 x xi Pr(X < xi) Pr(X xi) Pr(X = xi) Pr(X xi) Pr(X > xi) x1 0.56 0.8064 0.2464 0.44 0.1936 x2 0.8064 0.91482 0.10842 0.1936 0.08518 2005 Thomas Mitchell, All Rights Reserved r every trial, the probability ariable with a \"geometric\" ) = p (1 - p)x-1, x = 1, 2, 3, ... Cumulative distribution Probability: Pr(X = x) 6 17 18 19 20 21 22 23 24 25 26 27 28 29 30 v 1.1 / 03.22.05